Two Means - Unknown, Unequal Variance: Videos & Practice Problems

In hypothesis testing involving two samples, the primary focus shifts from a single mean to the difference between two sample means. The process remains fundamentally similar to that of a one-sample test, involving the formulation of hypotheses, calculation of test statistics, determination of p-values, and drawing conclusions.

The initial step is to establish the null hypothesis, which posits that the two means are equal: H_0: \mu_1 = \mu_2. This can also be expressed as H_0: \mu_1 - \mu_2 = 0. The alternative hypothesis, which indicates a difference, is typically framed as H_a: \mu_1 \neq \mu_2, suggesting a two-tailed test.

Before proceeding, it is essential to verify certain conditions: the samples must be random and independent, the population standard deviations (\sigma_1 and \sigma_2) are unknown and not assumed to be equal, and the samples should either be normally distributed or sufficiently large.

The test statistic for a two-sample t-test is calculated using the formula:

t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

In this formula, \bar{x}_1 and \bar{x}_2 represent the sample means, s_1 and s_2 are the sample standard deviations, and n_1 and n_2 are the sample sizes. The difference in population means is assumed to be zero under the null hypothesis.

After calculating the t-score, the next step is to determine the p-value, which indicates the probability of observing the test statistic under the null hypothesis. For two samples, the degrees of freedom can be approximated by taking the smaller of the two sample sizes minus one: df = min(n_1, n_2) - 1.

Once the p-value is obtained, it is compared to the significance level (alpha). If the p-value is less than alpha, the null hypothesis is rejected, indicating sufficient evidence to support the alternative hypothesis. For example, if the p-value is 0.0005 and alpha is set at 0.05, the conclusion would be to reject the null hypothesis, suggesting a significant difference in means.

In summary, conducting a two-sample hypothesis test involves establishing hypotheses about the means, ensuring the validity of assumptions, calculating the test statistic, determining the p-value, and making a conclusion based on the comparison with the significance level. This structured approach allows for a clear understanding of differences between groups, such as in studies comparing mean resting heart rates between males and females.

In hypothesis testing, particularly when comparing two population means, it is essential to follow a structured approach even when the problem lacks context. The first step involves formulating the null hypothesis (H₀) and the alternative hypothesis (H_a). In this scenario, the claim is that the mean of the first population (μ₁) is greater than the mean of the second population (μ₂). Therefore, the null hypothesis is that the means are equal (H₀: μ₁ = μ₂), while the alternative hypothesis is H_a: μ₁ > μ₂.

Next, we calculate the test statistic, which in this case is a t-score due to unknown and unequal population standard deviations. The formula for the t-score is given by:

t = \frac{\bar{x}_1 - \bar{x}_2 - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}

Here, \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means, \(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes. For example, if \(\bar{x}_1 = 462\), \(\bar{x}_2 = 431\), \(s_1 = 67\), \(n_1 = 32\), \(s_2 = 85\), and \(n_2 = 19\), the t-score can be calculated as follows:

t = \frac{462 - 431 - 0}{\sqrt{\frac{67^2}{32} + \frac{85^2}{19}}} = 1.359

After obtaining the t-score, the next step is to determine the p-value, which represents the probability of observing a t-score as extreme as the one calculated, under the null hypothesis. Since this is a right-tailed test, we look for the area to the right of the t-score in the t-distribution. The degrees of freedom can be approximated as the smaller of the two sample sizes minus one, which in this case is 18.

Using statistical software or a t-distribution table, we find that the p-value corresponding to a t-score of 1.359 with 18 degrees of freedom is approximately 0.0955. The final step in hypothesis testing is to compare the p-value to the significance level (α), which is set at 0.1 in this example. Since the p-value (0.0955) is less than α (0.1), we reject the null hypothesis.

This leads us to conclude that there is sufficient evidence to support the claim that μ₁ is greater than μ₂. In summary, through systematic steps of hypothesis testing, we can draw meaningful conclusions even from limited information.

When conducting a hypothesis test involving two samples, it's essential to understand how to construct a confidence interval for the difference in means. This process is similar to creating a confidence interval for a single mean, with some modifications to the point estimator and margin of error. The confidence interval will estimate the difference between the population means using the sample means.

To begin, ensure that the samples are random and independent. For example, if you are comparing the mean resting heart rates of males and females, you can assume independence since the samples do not interact. Next, check that the populations are normally distributed or that the sample sizes are sufficiently large. If the sample sizes are small, as long as you can assume normality, you can proceed.

The next step involves finding the critical value, denoted as \( t_{\alpha/2} \), which corresponds to your confidence level. For a 90% confidence level, you will look for the t-value that leaves 5% in each tail of the distribution. The degrees of freedom for this calculation is determined by the smaller sample size minus one. For instance, if the smaller sample size is 10, the degrees of freedom would be 9. Using a t-table, you would find that \( t_{0.05, 9} \approx 1.833 \).

Now, calculate the point estimator, which is the difference between the sample means, \( \bar{x}_1 - \bar{x}_2 \). For example, if the sample means are 70.2 and 81.4, the point estimator would be \( 70.2 - 81.4 = -11.2 \).

Next, compute the margin of error using the formula:

\[E = t_{\alpha/2} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]

Here, \( s_1 \) and \( s_2 \) are the sample standard deviations, and \( n_1 \) and \( n_2 \) are the sample sizes. For example, if \( s_1 = 5.8 \) and \( s_2 = 6.4 \) with sample sizes of 10 and 11 respectively, you would calculate the margin of error accordingly. After performing the calculations, you might find a margin of error of approximately 4.88.

Finally, construct the confidence interval by adding and subtracting the margin of error from the point estimator. For instance, the lower bound would be \( -11.2 - 4.88 = -16.08 \) and the upper bound would be \( -11.2 + 4.88 = -6.32 \). Thus, the 90% confidence interval for the difference in means would be \( (-16.08, -6.32) \).

This interval suggests that we are 90% confident that the true difference in mean resting heart rates between males and females lies between these two values. To evaluate a claim of no difference in means, check if the interval includes zero. Since both bounds are negative, zero is not included, indicating a significant difference. Therefore, you would reject the null hypothesis that there is no difference in mean resting heart rates between the two groups.